Solving the time-varying tensor square root equation by varying-parameters finite-time Zhang neural network

Abstract

We study the time-varying (TV) tensor square root problem, which could be regarded as a generalization of both the time-invariant (TI) tensor square root and the TV/TI matrix square root problems. The existence and uniqueness of the solution to the TV tensor square root problem are discussed. A new general varying-parameters finite-time convergent Zhang neural network model (VPsFTZNN) is proposed. We use the original ZNN and the new VPsFTZNN model as software solutions to the TV tensor square root problem. The convergence results of ZNN and VPsFTZNN dynamical systems are given. We prove that the VPsFTZNN model converges in a finite-time which is shorter than the finite time convergence of existing finite-time ZNN models. Also, we show that a superior convergence can be achieved if we use the power-sigmoid or smooth power-sigmoid activation functions instead of linear activation function under certain conditions. For comparison, some other models are also introduced and used to compute the square root of TV tensors. Numerical examples for TI and TV tensor/matrix square root problems are elaborated to confirm the theoretical results and comparison results with various model further illustrate the reliability and superiority of the proposed VPsFTZNN dynamics. Representative example on the application of this model for solving TV linear equations is also given.